Properties

Label 5202.2.a
Level $5202$
Weight $2$
Character orbit 5202.a
Rep. character $\chi_{5202}(1,\cdot)$
Character field $\Q$
Dimension $112$
Newform subspaces $50$
Sturm bound $1836$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5202.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 50 \)
Sturm bound: \(1836\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(7\), \(23\), \(47\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5202))\).

Total New Old
Modular forms 990 112 878
Cusp forms 847 112 735
Eisenstein series 143 0 143

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(13\)
\(+\)\(+\)\(-\)$-$\(9\)
\(+\)\(-\)\(+\)$-$\(18\)
\(+\)\(-\)\(-\)$+$\(16\)
\(-\)\(+\)\(+\)$-$\(13\)
\(-\)\(+\)\(-\)$+$\(9\)
\(-\)\(-\)\(+\)$+$\(14\)
\(-\)\(-\)\(-\)$-$\(20\)
Plus space\(+\)\(52\)
Minus space\(-\)\(60\)

Trace form

\( 112 q + 112 q^{4} - 6 q^{5} - 4 q^{7} + O(q^{10}) \) \( 112 q + 112 q^{4} - 6 q^{5} - 4 q^{7} - 2 q^{10} + 2 q^{11} - 4 q^{14} + 112 q^{16} - 4 q^{19} - 6 q^{20} - 2 q^{22} + 120 q^{25} + 8 q^{26} - 4 q^{28} - 14 q^{29} + 4 q^{31} + 4 q^{35} + 18 q^{37} - 4 q^{38} - 2 q^{40} + 12 q^{41} + 4 q^{43} + 2 q^{44} + 24 q^{46} + 12 q^{47} + 108 q^{49} - 16 q^{50} + 4 q^{53} + 48 q^{55} - 4 q^{56} + 6 q^{58} + 4 q^{59} + 26 q^{61} + 20 q^{62} + 112 q^{64} + 28 q^{65} + 12 q^{67} + 28 q^{70} + 24 q^{73} - 10 q^{74} - 4 q^{76} - 24 q^{77} - 8 q^{79} - 6 q^{80} + 4 q^{82} + 8 q^{83} + 24 q^{86} - 2 q^{88} - 40 q^{89} + 24 q^{91} + 12 q^{94} - 24 q^{95} + 20 q^{97} + 8 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5202))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 17
5202.2.a.a 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(-4\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-4q^{5}+3q^{7}-q^{8}+4q^{10}+\cdots\)
5202.2.a.b 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(-3\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-3q^{5}+4q^{7}-q^{8}+3q^{10}+\cdots\)
5202.2.a.c 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-4q^{11}+\cdots\)
5202.2.a.d 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(0\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+4q^{7}-q^{8}+6q^{11}+2q^{13}+\cdots\)
5202.2.a.e 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(3\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+3q^{5}-4q^{7}-q^{8}-3q^{10}+\cdots\)
5202.2.a.f 5202.a 1.a $1$ $41.538$ \(\Q\) None \(-1\) \(0\) \(4\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+4q^{5}-3q^{7}-q^{8}-4q^{10}+\cdots\)
5202.2.a.g 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(-4\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-4q^{5}+2q^{7}+q^{8}-4q^{10}+\cdots\)
5202.2.a.h 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(-2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{5}+2q^{7}+q^{8}-2q^{10}+\cdots\)
5202.2.a.i 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
5202.2.a.j 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(0\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-2q^{7}+q^{8}+2q^{13}-2q^{14}+\cdots\)
5202.2.a.k 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(0\) \(-1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{7}+q^{8}-q^{13}-q^{14}+\cdots\)
5202.2.a.l 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{7}+q^{8}-q^{13}+q^{14}+\cdots\)
5202.2.a.m 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
5202.2.a.n 5202.a 1.a $1$ $41.538$ \(\Q\) None \(1\) \(0\) \(2\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
5202.2.a.o 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(-4\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2+\beta )q^{5}+(2+2\beta )q^{7}+\cdots\)
5202.2.a.p 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{33}) \) None \(-2\) \(0\) \(-3\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1-\beta )q^{5}-\beta q^{7}-q^{8}+\cdots\)
5202.2.a.q 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}+(-2+\beta )q^{7}-q^{8}+\cdots\)
5202.2.a.r 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2\beta q^{5}-\beta q^{7}-q^{8}-2\beta q^{10}+\cdots\)
5202.2.a.s 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2\beta q^{5}-3\beta q^{7}-q^{8}+\cdots\)
5202.2.a.t 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}-3\beta q^{7}-q^{8}-\beta q^{10}+\cdots\)
5202.2.a.u 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}-q^{8}-\beta q^{10}-\beta q^{11}+\cdots\)
5202.2.a.v 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2\beta q^{5}+2\beta q^{7}-q^{8}+\cdots\)
5202.2.a.w 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{33}) \) None \(-2\) \(0\) \(3\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(1+\beta )q^{5}+\beta q^{7}-q^{8}+\cdots\)
5202.2.a.x 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(4\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(2+\beta )q^{5}+(-2+2\beta )q^{7}+\cdots\)
5202.2.a.y 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{33}) \) None \(2\) \(0\) \(-3\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-1-\beta )q^{5}+\beta q^{7}+q^{8}+\cdots\)
5202.2.a.z 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+(-2-\beta )q^{7}+q^{8}+\cdots\)
5202.2.a.ba 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2\beta q^{5}+\beta q^{7}+q^{8}+2\beta q^{10}+\cdots\)
5202.2.a.bb 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+q^{8}+\beta q^{10}-4\beta q^{11}+\cdots\)
5202.2.a.bc 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2\beta q^{5}+3\beta q^{7}+q^{8}+\cdots\)
5202.2.a.bd 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+3\beta q^{7}+q^{8}+\beta q^{10}+\cdots\)
5202.2.a.be 5202.a 1.a $2$ $41.538$ \(\Q(\sqrt{33}) \) None \(2\) \(0\) \(3\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1+\beta )q^{5}-\beta q^{7}+q^{8}+\cdots\)
5202.2.a.bf 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(-6\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2+\beta _{1})q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bg 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(-6\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2-\beta _{1})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bh 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(-6\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-2-\beta _{1})q^{5}+q^{7}-q^{8}+\cdots\)
5202.2.a.bi 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(6\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(2+\beta _{1})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bj 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(6\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(2+\beta _{1})q^{5}-q^{7}-q^{8}+\cdots\)
5202.2.a.bk 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(-3\) \(0\) \(6\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(2-\beta _{1})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bl 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(-6\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2-\beta _{1})q^{5}-q^{7}+q^{8}+\cdots\)
5202.2.a.bm 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(-6\) \(3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2+\beta _{1})q^{5}+(1+2\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bn 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(0\) \(-9\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-\beta _{1}-\beta _{2})q^{5}+(-3+\cdots)q^{7}+\cdots\)
5202.2.a.bo 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(0\) \(9\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2\beta _{1}+\beta _{2})q^{5}+(3+\beta _{2})q^{7}+\cdots\)
5202.2.a.bp 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(6\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(2-\beta _{1})q^{5}+(-1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bq 5202.a 1.a $3$ $41.538$ \(\Q(\zeta_{18})^+\) None \(3\) \(0\) \(6\) \(3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(2+\beta _{1})q^{5}+q^{7}+q^{8}+\cdots\)
5202.2.a.br 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(-4\) \(0\) \(0\) \(-8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(\beta _{2}+\beta _{3})q^{5}+(-2+\beta _{2}+\cdots)q^{7}+\cdots\)
5202.2.a.bs 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(-4\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta _{1}q^{5}+2\beta _{3}q^{7}-q^{8}+\cdots\)
5202.2.a.bt 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(-4\) \(0\) \(0\) \(8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-\beta _{2}+\beta _{3})q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)
5202.2.a.bu 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(4\) \(0\) \(-8\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(-2-\beta _{2}+\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
5202.2.a.bv 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(4\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta _{1}q^{5}-2\beta _{3}q^{7}+q^{8}+\cdots\)
5202.2.a.bw 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(4\) \(0\) \(0\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2\beta _{1}q^{5}+2\beta _{3}q^{7}+q^{8}+\cdots\)
5202.2.a.bx 5202.a 1.a $4$ $41.538$ \(\Q(\zeta_{16})^+\) None \(4\) \(0\) \(8\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(2+\beta _{2}+\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5202))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5202)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\)\(^{\oplus 2}\)